Electronics formulation

Petersson and coworkers have extended this two-electron formulation of asymptotic convergence to many-electron atoms. They note that the second-order MoUer-Plesset correlation energy for a many-electron system may be written as a sum of pair energies, each describing the energetic effect of the electron correlation between that pair of electrons ... 

The next step is the identification of the concept of chemical hardness, 17, with the second derivative of the energy with respect to the number of electrons, formulated by Parr and Pearson [14]... 

Higher-order derivatives with respect to external potential define xi(r, r1), Xi(r, r1, r"), etc., and their response with N define j(r, rJ), g2(r, r, / ), etc. This chain of derivatives is diagrammatically depicted in Figure 25.1 [22]. Thus, an exact one-electron formulation of all chemical responses (linear and nonlinear hardness, FF) in terms of Kohn-Sham orbital of the unperturbed system was derived [22b]. 

Fig. 3. Molecular orbitals used in the many-electron formulation of bridge-assisted electron transfer...

The preceding many-electron formulation is useful to separate the different contributions to T b, and could be easily improved to include the whole set of valence electrons of the system. However, it is not certain that this method is the most convenient for the effective calculation of the electronic factor. Actually, in the very few many-electron calculations that have been reported in the literature, Tjb was evaluated globally. These studies are summarized below. 

The first electronic formulation for the hydrogen bond was by Latimer and Rodebush (1920). For dimerized water molecules, they wrote... 

In the earlier sections of this chapter we reviewed the many-electron formulation of the symmetry-adapted perturbation theory of two-body interactions. As we saw, all physically important contributions to the potential could be identified and computed separately. We follow the same program for the three-body forces and discuss a triple perturbation theory for interactions in trimers. We show how the pure three-body effects can be separated out and give working equations for the components in terms of molecular integrals and linear and quadratic response functions. These formulas have a clear, partly classical, partly quantum mechanical interpretation. The exchange terms are also classified for the explicit orbital formulas we refer to Ref. (302). 

The aromatic nature of the [18]porphyrin-(2.1.0.1) derivative 3.135, as well as analogs 3.136, 3.137, and 3.144 (systems we will hereafter refer to as corrphycenes), was confirmed by the observation of sustained diamagnetic ring current effects in its H NMR spectrum. The UV-vis absorption spectrum (in benzene) is also consistent with the proposed aromaticity and with an 18-7t-electron formulation. Specifically, a... 

Consistent with its 22 n-electron formulation, the free-base form of isosmar-agdyrin (6.11b) appears to be aromatic, as judged from a preliminary H NMR spectral analysis. On the other hand, the protonated form (6.11a) appears to be non-aromatic, as evidenced by the lack of any diamagnetic ring-current effects in its H NMR spectrum. Based upon integration of the NMR signals, the lack of aromaticity for 6.11a appears to be the result of protonation at one of the meso-positions of the macrocycle. This protonation effect thus resembles that observed for the 18 7c-electron corroles discussed in Chapter 2. 

The H NMR spectrum of 7.80 was also recorded by Elix. While it was of insufficient quality to allow an extract assignment of structure vide supra), it did allow Elix to conclude that, as expected given its 4n n-electron formulation, compound 7.80 is not aromatic as judged by the absence of any substantial macrocyclic ring-current effects. This conclusion reached, it should be noted, that the formally 4n + 2 pentaoxa[30]pentaphyrin-(2.2.2.2.2) isomers also failed to display any kind of substantial ring-current effects. This led Elix to conclude that there is no fundamental difference in electronic character between large 4n and 4n + 2 7r-electron oxido-bridged annulenes. ... 

Magnetic susceptibility, XPS and ESR confirm the +5 oxidation state of U with configuration [74], and therefore the electronic formulation U Nb sOio. 

Species 7 contains a stabilized oxygen atom, and the parallel chemistry with the active form of cytochrome P-450 prompts the conclusion that it also contains stabilized atomic oxygen. We have argued elsewhere20,31 that the most reasonable electronic formulation for the active form of cytochrome P-450 is (RS)(por)Fe =O with an RS-Fe covalent bond and an Fe=O covalent double bond. 

In a slightly different formulation, instead of current density J(f)> the tunneling transition can be characterized by quantum fluxes between atoms of the medium, Ja6, which are called interatomic tunneling currents[2]. In one-electron formulation, the interatomic currents can be expressed as follows ... 

The expressions for interatomic currents in many-electron formulation, unfortunately, are not as simple as Eq. (5), because of the exchange and overlap effects. (To define an atom, one needs to deal with the nonorthogonality of atomic orbitals of its neighbours, which complicates the formalism [6].)... 

It has the property that/pp = —IPp when the orbital p is doubly occupied and/pp = —EAp when the orbital is empty. The value will be somewhere between these two extremes for active orbitals. Thus, we have for orbitals with occupation number one /pp = — j(IPp + EAp). This formulation is somewhat unbalanced and will favor systems with open shells, leading, for example, to somewhat low binding energies [52]. The problem is that one would like to separate the energy connected with excitation out from an orbital from that of excitation into the orbital. This cannot be done within a one-electron formulation of the zeroth order Hamiltonian. K. Dyall has suggested to use a two-electron operator for the active part [53], but this leads to a too complicated formalism and also breaks important orbital invariance properties (the result is, for... 

This prediction is a reasonable one for most cerium pnictides, namely CeP, CeAs, CeSb, and CeBi which, in fact, exhibit localized spin moments with an antiferromagnetic ordering of the 4/ electron remaining on each Ce [268]. CeN, however, is a metallic conductor with the corresponding magnetic properties and it only shows Pauli paramagnetism of the metallic electron gas, such that no local spin moment, characteristic for an unpaired electron, can be detected. This behavior leads to the possibility of an electronic formulation according to with one electron left in the conduction band, but... 

Early electronic formulations of benzene could not explain aromatic stability but did nevertheless produce new systematizations. The classical paper of Kermack and Robinson in which benzene was represented by the symbol (4), essentially a restatement of Thiele s structure, led Robinson i to the idea of the aromatic sextet. Electronic theory at the time recognized the importance of stable associations of electrons in pairs and in octets but provided no basis for the stability attributed to a group of six electrons. 

The MSe compounds with trivalent rare earth cations (all except M = Sm, Eu, and Yb) exhibit low room temperature resistivities and positive temperature coefficients of resistance and appear to be degenerate semiconductors, or metallic in nature. The high conductivity is believed to result from the excess valence electron (formulated as [1]) which does... 

The electronic formulation of the structure of the boron hydrides encounters a number of difficulties. The ordinary concepts of valence will not suffice to explain their stmcture this is shown by the fact that in the simplest hydride, diborane B2H6, which has 2x3 + 6= 12 electrons, as many bonds must be explained as are required for C2H6 which has two more (2x4 + 6 = 14) electrons available. Thus it is that any structural theory for these compounds requires new hypotheses. 

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